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Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy under contract DE-AC04-94AL85000.

Reconnect ‘04

Using PICO

Cynthia Phillips, Sandia National Labs

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Slide 2

AMPL Model for 1| prec |

• Declare/compute parameters, sets

param n > 0 integer; # Number of job

set Jobs := 0..n-1;

param p {Jobs} >= 0 integer; # Job Duration p(j) param w {Jobs} >= 0 integer; # Job weight (for objective) w(j)

# (predecessor, successor) pairs

set Precedence within Jobs cross Jobs;

# The makespan of the schedule (and latest finish time) param T := sum{j in Jobs} p[j];

w jC j

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Slide 3


• Declare variables

var x {j in Jobs, t in p[j]..T} binary;

w jC j

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Slide 4


minimize WCT:

sum{j in Jobs, t in p[j]..T} w[j] * t * x[j,t];

• subject to TaskDone {j in Jobs}:

sum{t in p[j]..T} x[j,t] = 1;• subject to UnitWork {tu in 1..T}:

sum{j in Jobs, t in tu..tu+p[j]-1 inter p[j]..T} x[j,t] <= 1;

• subject to PrecConstraint{j in Jobs,k in Jobs,

tu in p[j]..T-p[k]: (j,k) in Precedence}:sum{t in p[j]..tu}x[j,t] >= sum{t in p[j]+p[k]..tu + p[k]} x[k,t];

w jC j

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Slide 5


minimize WCT:

sum{j in Jobs, t in p[j]..T} w[j] * t * x[j,t];

Corresponds to:

subject to TaskDone {j in Jobs}:

sum{t in p[j]..T} x[j,t] = 1;

Corresponds to:

w jC j

x jt

t p j

T

1 j

min w j

t p j

T

j1

n

tx jt

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Slide 6


subject to UnitWork {tu in 1..T}:

sum{j in Jobs, t in tu..tu+p[j]-1 inter p[j]..T} x[j,t] <= 1;

Corresponds to:

subject to PrecConstraint{j in Jobs,k in Jobs,

tu in p[j]..T-p[k]: (j,k) in Precedence}:

sum{t in p[j]..tu}x[j,t] >= sum{t in p[j]+p[k]..tu + p[k]} x[k,t];

Corresponds to:

w jC j

x ju 1u t

t p j 1

j1

n

t 1,..., T p j j1

n

x ju

u1

t

xku

u1

t pk

0 J i J k , t 1,...,T pk

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Slide 7

More Complex Sets to Reduce IP size

# Transitive closure of precedence relations.set step{s in 1..ceil((n-1)/2)} dimen 2 :=

if s==1

then Precedence

else step[s-1] union setof {k in Jobs, (i,k) in step[s-1], (k,j) in step[s-1]} (i,j);

set TransPrec := step[ceil((n-1)/2)];

# Earliest finish time

param EFT{j in Jobs} := p[j] + sum{k in Jobs: (k,j) in TransPrec} p[k];

# Latest finish time

param LFT{j in Jobs} := T - sum{k in Jobs: (j,k) in TransPrec} p[k];# Possible finish times

set FinishWindow{j in Jobs} := EFT[j]..LFT[j];

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Slide 8

Smaller, simpler formulation

var x {j in Jobs, t in FinishWindow[j]} binary;minimize WCT:

sum{j in Jobs, t in FinishWindow[j]} w[j] * t * x[j,t];

subject to TaskDone {j in Jobs}:

sum{t in FinishWindow[j]} x[j,t] = 1;

subject to UnitWork {tu in 1..T}:

sum{j in Jobs, t in tu..tu+p[j]-1 inter FinishWindow[j]} x[j,t] <= 1;

[Similar changes for precedence constraints]

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Slide 9

AMPL Data Files - Straightforward

param n := 6; param p :=

0 8

1 2

2 1

3 3

4 12

5 1;

set Precedence :=

0 21 2

2 3

4 5;

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Slide 10

Use Sandia National Labs IP Solver: PICO

• Write cutting-plane and approximate-solution code using AMPL

variables

• Mapping transparent

Data

FilesSolver:

PICO

Exact

ComputeApproximate

Solution

IP

LPModel

Files

AMPL

Cutting Planes

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Slide 11

AMPL-PICO Interface

Standard AMPL interfaces

Customized PICO Interface

AMPL

SoftwareAMPL Model File

AMPL Problem

Specification Files

PICO

Executable

AMPL SolutionSpecification Files

AMPLSoftware

AMPL Solver Output

Unix ScriptsAMPL Model FileTailored PICO

C++ FilesC++ Compiler

Tailored PICO

ExecutablePICO Output

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Slide 12

Caveats: Status as of June 22, 2004

• The derived classes are for branch and bound only (no cuts yet; notstable; but they’ll be there “soon”)

• Only the serial version works as of this morning

– This could change by tomorrow

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Slide 13

Customized PICO Application

Goal: transparent access to AMPL variables from within problem-specificderivatives of core PICO application classes:

– MILP The “master” branching class

– MILPNode The “worker” subproblem class

– MILPProblem The problem definition class

• Creates a new namespace, so (as far as you’re concerned), these names

are the same.

• Maps multidimensional ampl variables to PICO’s linear variable space

– Foo(1,2)

– Foo(bar(stuff),callFunc(moreStuff))

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Slide 14

How PICO Learns AMPL Names

• Variable names, constraint names – Automatic (when the customized code is set up)

• Other parameters: list explicitly

– You may never use some helper parameters from the ampl file

# PICO SYMBOL: p n w EFT LFT T

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Slide 15

Example: Incumbent Heuristic for 1|prec|

Computing midpoints by stepping through valid x in lexicographic order

w jC j

DoubleVector frac(n());

IntVector time(n());

int curr_job = -1;

x_const_iterator curr = x_valid().begin();

x_const_iterator last = x_valid().end();

while (curr != last) {frac[curr->val1] += solution[x(*curr)];

if (frac[curr->val1] > 0.5) {

time[curr->val1] = curr->val2;

//

// Step past remaining jobs

//

curr_job = curr->val1;

while (curr->val1 == curr_job)

curr++;

}

else

curr++;

}

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Slide 16

Alternative Iteration

Computing midpoints

DoubleVector frac(n());

IntVector time(n());

for (int i=0;i< n(); i++){

for (int j=EFT(i) ; j<LFT(i) ; j++){

if (x_valid.isMember(i,j)){frac[i] += solution[x(i,j)];

if (frac[i] > 0.5) {

time[i] = j;

continue;

}

}

}

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Slide 18

Another Example: computing objective

DoubleVector soln(n()); IntVector index(n());

order(time,index); // sorting

soln [0] = p(index[0]);

value = p(index[0]) * w(index[0]);for (int i=1; i< n(); i++) {

soln[index[i]] = soln[index[i-1]] + p(index[i-1]);

value += soln[index[i]] * w(index[i]);

}

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Slide 19

Incumbent Heuristics

• These code fragments would be part of the incumbentHeuristic() methodin MILPNode.

• If you find an incumbent, this method must call

globalPtr updateIncumbent(solution, solution value)

This is a PICO (1D) solution.

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Slide 20

A Customized PICO Application: A Simple Example

Command line:gen_milp_app test.mod test.dat

Generates:

test.col Column labelstest.row Row labels

test.val Values specified as PICO SYMBOLs

test.mps The MPS file for this application

test.map A data file that describes the sets andsymbols defined by test.mod

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Slide 21

A Customized PICO Application: A Simple Example

Command line:gen_milp_app test.mod test.dat

Generates:

Makefile-test generates executableMakefile symbolic link to Makefile-test

test_info.{h,cpp} data structures to map ampl variables

test_milp.{h,cpp} derived classes for branching, nodes, etc.

test_extras.{h,cpp} for your custom methods (never deleted)

Where name.{h,cpp} means name.h and name.cpp

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Slide 22

Staged generation

• Stage 1 runs ampl to get row, column, val files, etc• Stage 2 generates the C++ files with derived milp classes

You can break this up:

gen_milp_app -stage 1 test.mod test.dat

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Slide 23

Syntactic Components in test.map

The AMPL/PICO interface dynamically identifies:• Sets: a collection a n-tuples.

– Sets of string literals: { aa, bb, cc, dd }

– Sets of integers: {1, 2, 3, 5, 7}

– Sets of n-tuples defined from other sets:

{(aa,1), (bb,2), (cc,3), (dd,5)}

• Symbols: any row or column label

Additional sets and data values are exported from AMPL by adding to

test.mod:

# PICO SYMBOLS: <symbol1> <symbol2> …

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Slide 24

Editing the map file

• Index sets are read from ampl row/col files, not model files – Cannot connect similar index sets

Set0(INT);

Set1(INT);

…. Set6(INT);

TaskDone[Set0];

UnitWork[Set1]

PrecConstraint[Set2, Set3, Set4];X[Set5, Set6];

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Slide 25

Editing the map file

• Clean up to reflect our knowledgeJobs(INT);

Time(INT);

TaskDone[Jobs];

UnitWork[Time]PrecConstraint[Jobs, Jobs, Time];

X[Jobs, Time];

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Slide 26

Protected Symbolic Access

Problem: the AMPL symbols may conflict with symbols in a the PICOclasses.

Solution:

• Access AMPL symbols only through the info() methodUse info p(j) instead of p(j)

• Enable protection by

– Compiling with the –DPROTECTED_APPINFO flag

– Running gen_milp_app with the–

protected-vars flag

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Slide 27

Using the Customized PICO Classes

By default gen_milp_app generates a generic main() function(parallel if you have configured with mpi, serial otherwise)

The following commands generate, compile and execute the PICO classes

gen_milp_app test.mod test.datmake all

test_milp test.mps

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Slide 28

Overriding Parameters

All parameters have default values. test_milp --help

Should output information about parameters you can override

test_milp --debug=50 --useSets --preprocessIP=0 test.mps

If you don’t specify a value during the override, PICO will assume 1

• Fine for binary parameters

• Parameters can have any type (probably effectively any built-in type)

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Slide 29

Generating Cutting Planes

• Conforms to Open Source Initiative (OSI) conventions

• Uses (and will contribute to) Cut Generation Library (CGL)

• Cuts are sparse (you have to build them)

Cut Generator

PICO LPinterface

LP Solution

picoRowCut

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Slide 30

ACRO

• acro-pico gives most of the pieces you need: pico, utilib, soplex, COIN• Configuration facility

• For this workshop, you can check out from a local cvs repository

• Example configuration

Sets up makfiles that link properly to local MPI libraries, cplex, etc

Configure --site=lafayette --without-mpi --with-clp --without-cplex --with-debugging

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Slide 31

Comparison with ABACUS

A Branch And CUt SystemVariables and Constraints are C++ classes.

Variables must implement

double coeff(CONSTRAINT *c);

Constraints must implementdouble coeff(VARIABLE *v);

Flexible, reasonably quick development, but

• User manages mapping of individual variables to linear ordering• Cutting planes are accessed densely

• Have to compute the coefficients of all constraints, not just cuts

Documents

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